3.2215 \(\int \frac{f+g x}{(d+e x)^4 \sqrt{c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\)

Optimal. Leaf size=285 \[ -\frac{16 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x) (2 c d-b e)^4}-\frac{8 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x)^2 (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{35 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (d+e x)^4 (2 c d-b e)} \]

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*e^2*(2*c*d - b*e)*
(d + e*x)^4) - (2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2])/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^3) - (8*c*(6*c*e*f + 8*c*d*g - 7*b*
e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*
x)^2) - (16*c^2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2])/(105*e^2*(2*c*d - b*e)^4*(d + e*x))

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Rubi [A]  time = 1.00305, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.068 \[ -\frac{16 c^2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x) (2 c d-b e)^4}-\frac{8 c \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{105 e^2 (d+e x)^2 (2 c d-b e)^3}-\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-7 b e g+8 c d g+6 c e f)}{35 e^2 (d+e x)^3 (2 c d-b e)^2}-\frac{2 (e f-d g) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{7 e^2 (d+e x)^4 (2 c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)/((d + e*x)^4*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]

[Out]

(-2*(e*f - d*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(7*e^2*(2*c*d - b*e)*
(d + e*x)^4) - (2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c
*e^2*x^2])/(35*e^2*(2*c*d - b*e)^2*(d + e*x)^3) - (8*c*(6*c*e*f + 8*c*d*g - 7*b*
e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(105*e^2*(2*c*d - b*e)^3*(d + e*
x)^2) - (16*c^2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e
^2*x^2])/(105*e^2*(2*c*d - b*e)^4*(d + e*x))

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Rubi in Sympy [A]  time = 115.041, size = 272, normalized size = 0.95 \[ \frac{16 c^{2} \left (7 b e g - 8 c d g - 6 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{105 e^{2} \left (d + e x\right ) \left (b e - 2 c d\right )^{4}} - \frac{8 c \left (7 b e g - 8 c d g - 6 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{105 e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )^{3}} + \frac{2 \left (7 b e g - 8 c d g - 6 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{35 e^{2} \left (d + e x\right )^{3} \left (b e - 2 c d\right )^{2}} - \frac{2 \left (d g - e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{7 e^{2} \left (d + e x\right )^{4} \left (b e - 2 c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)/(e*x+d)**4/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

16*c**2*(7*b*e*g - 8*c*d*g - 6*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c
*d))/(105*e**2*(d + e*x)*(b*e - 2*c*d)**4) - 8*c*(7*b*e*g - 8*c*d*g - 6*c*e*f)*s
qrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e + c*d))/(105*e**2*(d + e*x)**2*(b*e - 2*c*
d)**3) + 2*(7*b*e*g - 8*c*d*g - 6*c*e*f)*sqrt(-b*e**2*x - c*e**2*x**2 + d*(-b*e
+ c*d))/(35*e**2*(d + e*x)**3*(b*e - 2*c*d)**2) - 2*(d*g - e*f)*sqrt(-b*e**2*x -
c*e**2*x**2 + d*(-b*e + c*d))/(7*e**2*(d + e*x)**4*(b*e - 2*c*d))

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Mathematica [A]  time = 0.72759, size = 163, normalized size = 0.57 \[ \frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} \left (-8 c^2 (d+e x)^3 (-7 b e g+8 c d g+6 c e f)-3 (d+e x) (b e-2 c d)^2 (-7 b e g+8 c d g+6 c e f)+4 c (d+e x)^2 (b e-2 c d) (-7 b e g+8 c d g+6 c e f)+15 (b e-2 c d)^3 (e f-d g)\right )}{105 e^2 (d+e x)^4 (b e-2 c d)^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(f + g*x)/((d + e*x)^4*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2]),x]

[Out]

(2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(15*(-2*c*d + b*e)^3*(e*f - d*g) - 3*(
-2*c*d + b*e)^2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*(d + e*x) + 4*c*(-2*c*d + b*e)*(6*
c*e*f + 8*c*d*g - 7*b*e*g)*(d + e*x)^2 - 8*c^2*(6*c*e*f + 8*c*d*g - 7*b*e*g)*(d
+ e*x)^3))/(105*e^2*(-2*c*d + b*e)^4*(d + e*x)^4)

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Maple [A]  time = 0.018, size = 382, normalized size = 1.3 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 56\,b{c}^{2}{e}^{4}g{x}^{3}-64\,{c}^{3}d{e}^{3}g{x}^{3}-48\,{c}^{3}{e}^{4}f{x}^{3}-28\,{b}^{2}c{e}^{4}g{x}^{2}+256\,b{c}^{2}d{e}^{3}g{x}^{2}+24\,b{c}^{2}{e}^{4}f{x}^{2}-256\,{c}^{3}{d}^{2}{e}^{2}g{x}^{2}-192\,{c}^{3}d{e}^{3}f{x}^{2}+21\,{b}^{3}{e}^{4}gx-164\,{b}^{2}cd{e}^{3}gx-18\,{b}^{2}c{e}^{4}fx+524\,b{c}^{2}{d}^{2}{e}^{2}gx+120\,b{c}^{2}d{e}^{3}fx-416\,{c}^{3}{d}^{3}egx-312\,{c}^{3}{d}^{2}{e}^{2}fx+6\,{b}^{3}d{e}^{3}g+15\,{b}^{3}{e}^{4}f-46\,{b}^{2}c{d}^{2}{e}^{2}g-108\,{b}^{2}cd{e}^{3}f+144\,b{c}^{2}{d}^{3}eg+276\,b{c}^{2}{d}^{2}{e}^{2}f-104\,{c}^{3}{d}^{4}g-288\,{c}^{3}{d}^{3}ef \right ) }{105\, \left ({b}^{4}{e}^{4}-8\,{b}^{3}cd{e}^{3}+24\,{b}^{2}{c}^{2}{d}^{2}{e}^{2}-32\,b{c}^{3}{d}^{3}e+16\,{c}^{4}{d}^{4} \right ){e}^{2} \left ( ex+d \right ) ^{3}}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)/(e*x+d)^4/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)

[Out]

-2/105*(c*e*x+b*e-c*d)*(56*b*c^2*e^4*g*x^3-64*c^3*d*e^3*g*x^3-48*c^3*e^4*f*x^3-2
8*b^2*c*e^4*g*x^2+256*b*c^2*d*e^3*g*x^2+24*b*c^2*e^4*f*x^2-256*c^3*d^2*e^2*g*x^2
-192*c^3*d*e^3*f*x^2+21*b^3*e^4*g*x-164*b^2*c*d*e^3*g*x-18*b^2*c*e^4*f*x+524*b*c
^2*d^2*e^2*g*x+120*b*c^2*d*e^3*f*x-416*c^3*d^3*e*g*x-312*c^3*d^2*e^2*f*x+6*b^3*d
*e^3*g+15*b^3*e^4*f-46*b^2*c*d^2*e^2*g-108*b^2*c*d*e^3*f+144*b*c^2*d^3*e*g+276*b
*c^2*d^2*e^2*f-104*c^3*d^4*g-288*c^3*d^3*e*f)/(e*x+d)^3/e^2/(b^4*e^4-8*b^3*c*d*e
^3+24*b^2*c^2*d^2*e^2-32*b*c^3*d^3*e+16*c^4*d^4)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2
)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.41856, size = 818, normalized size = 2.87 \[ -\frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (8 \,{\left (6 \, c^{3} e^{4} f +{\left (8 \, c^{3} d e^{3} - 7 \, b c^{2} e^{4}\right )} g\right )} x^{3} + 4 \,{\left (6 \,{\left (8 \, c^{3} d e^{3} - b c^{2} e^{4}\right )} f +{\left (64 \, c^{3} d^{2} e^{2} - 64 \, b c^{2} d e^{3} + 7 \, b^{2} c e^{4}\right )} g\right )} x^{2} + 3 \,{\left (96 \, c^{3} d^{3} e - 92 \, b c^{2} d^{2} e^{2} + 36 \, b^{2} c d e^{3} - 5 \, b^{3} e^{4}\right )} f + 2 \,{\left (52 \, c^{3} d^{4} - 72 \, b c^{2} d^{3} e + 23 \, b^{2} c d^{2} e^{2} - 3 \, b^{3} d e^{3}\right )} g +{\left (6 \,{\left (52 \, c^{3} d^{2} e^{2} - 20 \, b c^{2} d e^{3} + 3 \, b^{2} c e^{4}\right )} f +{\left (416 \, c^{3} d^{3} e - 524 \, b c^{2} d^{2} e^{2} + 164 \, b^{2} c d e^{3} - 21 \, b^{3} e^{4}\right )} g\right )} x\right )}}{105 \,{\left (16 \, c^{4} d^{8} e^{2} - 32 \, b c^{3} d^{7} e^{3} + 24 \, b^{2} c^{2} d^{6} e^{4} - 8 \, b^{3} c d^{5} e^{5} + b^{4} d^{4} e^{6} +{\left (16 \, c^{4} d^{4} e^{6} - 32 \, b c^{3} d^{3} e^{7} + 24 \, b^{2} c^{2} d^{2} e^{8} - 8 \, b^{3} c d e^{9} + b^{4} e^{10}\right )} x^{4} + 4 \,{\left (16 \, c^{4} d^{5} e^{5} - 32 \, b c^{3} d^{4} e^{6} + 24 \, b^{2} c^{2} d^{3} e^{7} - 8 \, b^{3} c d^{2} e^{8} + b^{4} d e^{9}\right )} x^{3} + 6 \,{\left (16 \, c^{4} d^{6} e^{4} - 32 \, b c^{3} d^{5} e^{5} + 24 \, b^{2} c^{2} d^{4} e^{6} - 8 \, b^{3} c d^{3} e^{7} + b^{4} d^{2} e^{8}\right )} x^{2} + 4 \,{\left (16 \, c^{4} d^{7} e^{3} - 32 \, b c^{3} d^{6} e^{4} + 24 \, b^{2} c^{2} d^{5} e^{5} - 8 \, b^{3} c d^{4} e^{6} + b^{4} d^{3} e^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^4),x, algorithm="fricas")

[Out]

-2/105*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(8*(6*c^3*e^4*f + (8*c^3*d*e^3
- 7*b*c^2*e^4)*g)*x^3 + 4*(6*(8*c^3*d*e^3 - b*c^2*e^4)*f + (64*c^3*d^2*e^2 - 64
*b*c^2*d*e^3 + 7*b^2*c*e^4)*g)*x^2 + 3*(96*c^3*d^3*e - 92*b*c^2*d^2*e^2 + 36*b^2
*c*d*e^3 - 5*b^3*e^4)*f + 2*(52*c^3*d^4 - 72*b*c^2*d^3*e + 23*b^2*c*d^2*e^2 - 3*
b^3*d*e^3)*g + (6*(52*c^3*d^2*e^2 - 20*b*c^2*d*e^3 + 3*b^2*c*e^4)*f + (416*c^3*d
^3*e - 524*b*c^2*d^2*e^2 + 164*b^2*c*d*e^3 - 21*b^3*e^4)*g)*x)/(16*c^4*d^8*e^2 -
32*b*c^3*d^7*e^3 + 24*b^2*c^2*d^6*e^4 - 8*b^3*c*d^5*e^5 + b^4*d^4*e^6 + (16*c^4
*d^4*e^6 - 32*b*c^3*d^3*e^7 + 24*b^2*c^2*d^2*e^8 - 8*b^3*c*d*e^9 + b^4*e^10)*x^4
+ 4*(16*c^4*d^5*e^5 - 32*b*c^3*d^4*e^6 + 24*b^2*c^2*d^3*e^7 - 8*b^3*c*d^2*e^8 +
b^4*d*e^9)*x^3 + 6*(16*c^4*d^6*e^4 - 32*b*c^3*d^5*e^5 + 24*b^2*c^2*d^4*e^6 - 8*
b^3*c*d^3*e^7 + b^4*d^2*e^8)*x^2 + 4*(16*c^4*d^7*e^3 - 32*b*c^3*d^6*e^4 + 24*b^2
*c^2*d^5*e^5 - 8*b^3*c*d^4*e^6 + b^4*d^3*e^7)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{f + g x}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)/(e*x+d)**4/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)

[Out]

Integral((f + g*x)/(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(d + e*x)**4), x)

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GIAC/XCAS [A]  time = 0.732069, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(e*x + d)^4),x, algorithm="giac")

[Out]

sage0*x